MATH 583: Computing With Modular Forms, Spring 2006, UW.

Lecture Plan

[x] (Fri Apr 7) Level 1 modular forms 3: Structure theorem, Victor Miller Basis

[x] SAGE 1: Software for Algebra and Geoemtry Experiementation -- intro
[x] Level 1 modular forms 4: Hecke Operators; Maeda, Edixhoven
[x] Special Topic: Fast computation of Bernoulli numbers

[x] Modular symbols 1: basic definitions
[x] Modular symbols 2: modular symbols
[x] Modular symbols 3: Hecke operators

[x] Modular symbols 4: computing modular forms using modular symbols
[x] Dirichlet characters: intro and algorithm
[x] (April 28) C. Doran -- elliptic modular surfaces

[x] Dirichlet characters: more
[x] Movie -- "The Proof"
[x] Eisenstein series: computing an explicit basis

[x] Dimension formulas: statement and how to compute
[x] Linear algebra 1: computing echelon forms
[x] Linear algebra 2: decomposition algorithms

[x] Higher Weight Modular Symbols 1: basic definitions; how to compute:
             Sections 8.1 -- 8.2
[x] Higher Weight Modular Symbols 2: Hecke operators on them
             Sections 8.3

[x] Higher Weight Modular Symbols 3: the integration pairing
             Sections 8.5

[x] Newforms 1: Atkin-Lehner-Li theory
             Sections 9.1 -- 9.3
[x] Newforms 2: Computing (and storing!) systems of eigenvalues
             Section 9.4
[] Special Values of L-functions using modular symbols
             Sections 10.1-10.4

[] (may 29 -- memorial day holiday)
[] Enumeration of all elliptic curves of given conductor (Cremona's program):
             Sections 10.6-10.7
[] Sturm's bound: Congruences between modular forms; gen. Hecke algebras
             Chapter 11